Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 78691, 8 pages
doi:10.1155/2007/78691
Research Article
One Method for Proving Inequalities by Computer
Branko J. Malešević
Received 31 August 2006; Revised 30 October 2006; Accepted 31 October 2006
Recommended by Andrea Laforgia
We consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple
computer program.
Copyright © 2007 Branko J. Malešević. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Some particular inequalities
In this section we prove two new inequalities given in Theorems 1.2 and 1.10. While
proving these theorems we use a method for inequalities of the form f (x) ≥ 0, for the
continuous function f : [a,b] → R.
1.1. Let us consider some inequalities for the gamma function which is defined by the
integral
Γ(z) =
∞
0
e−t t z−1 dt
(1.1)
which converges for Re(z) > 0. In [1], the following statement is proved.
Lemma 1.1. For x ∈ [0,1] the following inequalities are true:
7
9
Γ(x + 1) < x2 − x + ,
4
5
9
(x + 2)Γ(x + 1) > .
5
(1.2)
2
Journal of Inequalities and Applications
The previous statement [1, Lemma 4.1] is proved by the approximative formula for
the gamma function Γ(x + 1) by the polynomial of the fifth order:
P5 (x) = −0.1010678x5 + 0.4245549x4 − 0.6998588x3 + 0.9512363x2 − 0.5748646x + 1
(1.3)
−5
which has the numerical bound of the absolute error ε = 5 · 10 for values of argument
x ∈ [0,1] [2, Formula 6.1.35, page 257].
In the Maple computer program we use numapprox package [3] for obtaining the
minimax rational approximation R(x) = Pm (x)/Qn (x) of the continuous function f (x)
over segment [a,b] (m is the degree of the polynomial Pm (x) and n is the degree of the
polynomial Qn (x)). Let ε(x) = f (x) − R(x) be the error function of an approximation
over segment [a,b]. Numerical computation of R(x) is given by Maple command
R := minimax f (x), x = a..b,[m,n],“err ” .
(1.4)
The result of the previous command is the minimax rational approximation R(x) and an
estimate for the value of the minimax norm of ε(x) as the number err (computation is
realized without the weight function). With the Maple minimax command a realization
of the Remez algorithm is given [4]. If it is not possible to determine minimax approximation in Maple program, there appears a message that it is necessary to increase decimal
degrees.
Let us assume that for the function f (x) the minimax rational approximation R(x) is
determined. Then, in the Maple the same estimate for the minimax norm of the error
function ε(x) is given by the command
err := infnorm ε(x), x = a..b .
(1.5)
The result of the previous command is number err := maxx∈[a,b] | f (x) − R(x)|. Practically
for the bound of the absolute error function |ε(x)| we use ε = err. Let us remark that the
bound of the absolute error ε is a numerical bound in the sense of [5] (approximation
errors on page 4), see also [6, 7].
Let us notice, as it is emphasized by [1, Remark 4.2], that for the proof of Lemma 1.1,
it is possible to use other polynomial approximations (of lower degree) of the functions
Γ(x + 1/2) and Γ(x + 1) for values x ∈ [0,1]. That idea is implemented in the next statement for Kurepa’s function which is defined by the integral
K(z) =
∞
0
e −t
tz − 1
dt,
t−1
(1.6)
which converges for Re(z) > 0 [8]. It is possible to make an analytical continuation of
Kurepa’s function K(z) to the meromorphic function with simple poles at z = −1 and z =
−n (n ≥ 3). Practically for computation values of Kurepa’s function we use the following
formula:
K(z) =
Ei(1) + iπ (−1)z Γ(1 + z)Γ(−z, −1)
+
e
e
(1.7)
Branko J. Malešević 3
which is cited in [9]. In the previous formula Ei(z) and Γ(z,a) are the exponential integral
and the incomplete gamma function, respectively. Let us numerically prove the following
statement.
Theorem 1.2. For x ∈ [0,1], the following inequality is true:
K(x) ≤ K (0)x,
(1.8)
where K (0) = 1.432205735 ... is the best possible constant.
Proof. Let us define the function f (x) = K (0)x − K(x) for x ∈ [0,1]. Let us prove f (x) ≥
0 for x ∈ [0,1]. Let us consider the continuous function
⎧
⎪
⎪
⎨α
:
x = 0,
⎪
⎩
:
x ∈ (0,1],
g(x) = ⎪ f (x)
x2
(1.9)
for constant α = −K (0)/2. Let us notice that the constant
α=−
f (x)
K (0)
K (0) − K (x)
K (0)x − K(x)
= lim
= lim
= lim
2
x
→
0+
x
→
0+
x
→
0+
2
2x
x
x2
(1.10)
is determined in the sense that g(x) is a continuous function over segment [0,1]. The
numerical value of that constant is
α=−
1
2
∞
0
e −t
log2 t
dt = 0.963321189 ...(> 0).
t−1
(1.11)
Using Maple we determine the minimax rational approximation for the function g(x) by
the polynomial of the first order:
P1 (x) = −0.531115454 x + 0.921004887
(1.12)
which has the bound of the absolute error ε1 = 0.04232 for values x ∈ [0,1]. The following
is true:
g(x) − P1 (x) − ε1 ≥ 0,
P1 (x) − ε1 > 0,
(1.13)
for values x ∈ [0,1]. Hence, for x ∈ [0,1] it is true that g(x) > 0 and f (x) ≥ 0 as well. Remark 1.3. Numerical values of constants K (0) and K (0) are determined by Maple
program. The numerical value of K (0) was first determined by Slavić in [10].
Corollary 1.4. Petojević in [11] used an auxiliary result K(x) ≤ 9/5x, for values x ∈ [0,1],
from [1, Lemma 4.3], for proving new inequalities for Kurepa’s function. Based on the previous theorem, all appropriate inequalities from [11] can be improved with a simple change of
fraction 9/5 with constant K (0).
4
Journal of Inequalities and Applications
1.2. Mitrinović and Vasić considered in [12] the lower bound of the arc sin-function,
which belongs to Shafer. Namely, the following statement is true.
Theorem 1.5. For 0 ≤ x ≤ 1 the following inequalities are true:
√
√
6 1+x− 1−x
3x
√
√
≤
≤ arc sin x.
2 + 1 − x2 4 + 1 + x + 1 − x
√
(1.14)
Fink proved the following statement in paper [13].
Theorem 1.6. For 0 ≤ x ≤ 1, the following inequalities are true:
3x
πx
√
≤ arc sin x ≤
.
2
2+ 1−x
2 + 1 − x2
√
(1.15)
Malešević proved the following statement in [14].
Theorem 1.7. For 0 ≤ x ≤ 1, the following inequalities are true:
π/(π − 2) x
3x
πx
√
√
≤ arc sin x ≤
≤
.
2 + 1 − x2
2/(π − 2) + 1 − x2 2 + 1 − x2
√
(1.16)
Remark 1.8. The upper bound of the arc sin-function
π/(π − 2) x
√
φ(x) =
2/(π − 2) + 1 − x2
(1.17)
is determined in paper [14] by λ-method of Mitrinović and Vasić [12].
Zhu proved the following statement in [15].
Theorem 1.9. For x ∈ [0,1], the following inequalities are true:
√
√
6 1+x− 1−x
3x
√
√
√
≤
≤ arc sin x
2
4+ 1+x+ 1−x
2+ 1−x
≤
π
√
√
√
2 + 1/2
1+x− 1−x
πx
√
√
√
≤
.
4+ 1+x+ 1−x
2 + 1 − x2
(1.18)
In this paper we give an improved statement of Zhu. Let us numerically prove the
following statement.
Theorem 1.10. For x ∈ [0,1], the following inequalities are true:
√
√
6 1+x− 1−x
3x
√
√
≤
≤ arc sin x
2 + 1 − x2 4 + 1 + x + 1 − x
√
√ √ √
√
π 2− 2 π −2 2
1+x− 1−x
√ √
√
≤ √
2(4 − π) π − 2 2 + 1 + x + 1 − x
√
√
√
π 2 + 1/2
1+x− 1−x
πx
√
√
√
≤
≤
.
4+ 1+x+ 1−x
2 + 1 − x2
(1.19)
Branko J. Malešević 5
Proof. Inequality
π
√
√
√
√ √ √
√
2 + 1/2
1+x− 1−x
π(2 − 2) π − 2 2
1+x− 1−x
√
√
√ √
√
≥ √
,
4+ 1+x+ 1−x
2(4 − π) π − 2 2 + 1 + x + 1 − x
(1.20)
for x ∈ [0,1], is directly verifiable by algebraic manipulations. Let us define the following
function:
√ √ √
√
π 2− 2 π −2 2
1+x− 1−x
√ √
√
− arc sin x,
f (x) = √
2(4 − π) π − 2 2 + 1 + x + 1 − x
(1.21)
for x ∈ [0,1]. Let us prove f (x) ≥ 0 for x ∈ [0,1], that is, f (sint) ≥ 0 for t ∈ [0,π/2]. Let
us define the function
⎧
⎪
⎪
⎪α
⎪
⎪
⎪
⎨ f (sint)
g(t) = ⎪ t 3 (π/2 − t)
⎪
⎪
⎪
⎪
⎪
⎩β
:
t = 0,
:
t ∈ 0,
:
t=
π
,
2
(1.22)
π
,
2
where α and β are constants determined with limits:
√ √
f (sint)
4 + 2 π − 12 2
√ = > 0,
α = lim 3
t →0+ t (π/2 − t)
24 − 12 2 π 2
√
√ √
(1.23)
f (sint)
16 2 − 16 + 8 − 4 2 π − 2π 2
√
β = lim 3
=
> 0.
t →π/2− t (π/2 − t)
2 2 − 2 π3
The previously determined function g(t) is continuous over [0,π/2]. Using Maple we
determine the minimax rational approximation for the function g(t) by the polynomial
of the first order:
P1 (t) = 0.000410754 t + 0.000543606
(1.24)
which has the bound of the absolute error ε1 = 1.408 · 10−5 for values t ∈ [0,π/2]. It is
true that
g(t) − P1 (t) − ε1 ≥ 0,
P1 (t) − ε1 > 0,
(1.25)
for values t ∈ [0,π/2]. Hence, for t ∈ [0,π/2] it is true that g(t)>0 and therefore f (sint) ≥
0 for t ∈ [0,π/2]. Finally f (x) ≥ 0 for x ∈ [0,1].
Remark 1.11. The paper [16] considers the upper bound of the arc sin-function
√ √ √
√
π 2− 2 π −2 2
1+x− 1−x
√ √
√
ϕ(x) = √
2(4 − π) π − 2 2 + 1 + x + 1 − x
via λ-method of Mitrinović and Vasić [12].
(1.26)
6
Journal of Inequalities and Applications
2. A method for proving inequalities
In this section we expose a numerical method for proving inequalities in following form:
f (x) ≥ 0
(2.1)
for the continuous function f : [a,b] → R. Let us assume that x = a is the root of the order
n and x = b is the root of the order m of the function f (x) (if x = a is not the root, then
we determine that n = 0, that is, if x = b is not the root, then we determine that m = 0).
The method is based on the first assumption that there exist finite and nonzero limits:
α = lim
x→a+
f (x)
,
(x − a)n (b − x)m
β = lim
x→b−
f (x)
.
(x − a)n (b − x)m
(2.2)
If for the function f (x) (over extended domain of [a,b]) at the point x = a there is an
approximation of the function by Taylor polynomial of nth order and at point x = b there
is an approximation of the function by Taylor polynomial of mth order, then
α=
f (n) (a)
,
n!(b − a)m
β = (−1)m
f (m) (b)
.
m!(b − a)n
(2.3)
Let us define the function
f
⎧
⎪
⎪
⎪α
⎪
⎪
⎪
⎨
f (x)
⎪
(x − a)n (b − x)m
⎪
⎪
g(x) = ga,b (x) = ⎪
⎪
⎪
⎩β
:
x = a,
:
x ∈ (a,b),
:
x=b
(2.4)
which is continuous over segment [a,b]. For proving inequality (2.1) we use the equivalence
f (x) ≥ 0 ⇐⇒ g(x) ≥ 0,
(2.5)
which is true for all values x ∈ [a,b]. Thus if α < 0 or β < 0, the inequality (2.1) is not true.
Hence, we consider only the cases α > 0 and β > 0. Let us notice that if the function f (x)
has only roots at some end-points of the segment [a,b], then (2.5) becomes f (x) ≥ 0 if
and only if g(x) > 0 for x ∈ [a,b]. The second assumption of the method is that there is
the minimax (polynomial) rational approximation R(x) = Pm (x)/Qn (x) of the function
g(x) over [a,b], which has the bound of the absolute error ε > 0 such that
R(x) − ε > 0,
(2.6)
for x ∈ [a,b]. Then g(x) > 0, for x ∈ [a,b]. Finally, on the basis (2.5), we can conclude
that f (x) ≥ 0, for x ∈ [a,b].
Let us emphasize that the minimax (polynomial) rational approximation of the function g(x) over [a,b] can be computed by Remez algorithm (via Maple minimax function
[3]). For applying Remez algorithm to the function g(x), it is sufficient that the function
Branko J. Malešević 7
is continuous. If g(x) is a differentiable function, then the second Remez algorithm is applicable [17]. According to the previous consideration, the problem of proving inequality
(2.1), in some cases, becomes a problem of existence of the minimax (polynomial) raf
tional approximation R(x) for g = ga,b (x) function with the bound of the absolute error
ε > 0 such that (2.6) is true. Let us notice that the problem of verification of inequality
(2.6) reduces to boolean combination of the polynomial inequalities.
Let us consider practical usages of the previously described method. For the function
of one variable the previous method can be applied to the inequality of the infinite interval using the appropriate substitute variable, which transforms inequality to the new one
over the finite interval. Next, if some limits in (2.2) are infinite, then, in some cases, the
initial inequality can be transformed, by the means of appropriate substitute variable, to
the case when the both limits in (2.2) are finite and nonzero.
The advantage of the described method is that for the function f (x) we do not have to
use some regularities concerning derivatives. Besides, the present method enables us to
obtain computer-assisted proofs of appropriate inequalities, which have been published
in journals which consider these topics. With this method we have obtained numerical
proofs of the appropriate inequalities from the following articles [18–28].
Finally, let us emphasize that the mentioned method can be extended and applied to
inequalities for multivariate functions by the means of appropriate multivariate minimax
rational approximations.
Acknowledgment
The research was partially supported by the MNTRS, Serbia, Grant no. 144020.
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Branko J. Malešević: Faculty of Electrical Engineering, University of Belgrade, P.O. Box 35-54,
11120 Belgrade, Serbia
Email addresses: malesh@eunet.yu; malesevic@etf.bg.ac.yu